fgm (version 1.0)

fgm: Functional Gaussian Graphical Model

Description

Estimates a sparse adjacency matrix representing the conditional dependency structure between features of a multivariate Gaussian process

Usage

fgm(y, L, alpha, gamma, t = seq(0, 1, length.out = dim(y[[1]])[2]),
  thr.FVE = 95, include.Omega = FALSE)

Arguments

y

list of length p containing densely observed multivariate (p-dimensional) functional data. y[[j]] is an nxm matrix of functional data for n subjects observed on a grid of length m

L

the number of eigenfunctions used for dimension reduction using the partially separable Karhunen-Loeve (PSKL) expansion obtained using `pfpca()`. This argument can take positive integer values greater or equal to 1.

alpha

penalty parameter for the common sparsity pattern taking values in [0,1].

gamma

penalty parameter for the overall sparsity pattern taking positive values.

t

(optional) grid on which functional data is observed, defaults to seq(0, 1, m) where m = dim(data[[1]])[2].

thr.FVE

this parameter sets a threshold for the minimum percentage of functional variance explained (FVE) by the PSKL eigenfunctions (obtained using `pfpca()`). This criterion is used only if a value for L is not provided or is greater than the maximum possible number of eigenfunctions estimated from y using pfpca().

include.Omega

logical variable indicating wheter to include the list of precision matrices in the output. Default value is FALSE.

Value

A list with letters and numbers.

A

Resulting adjacency matrix as the union of all the Omega matrices

L

number of PSKL expansion eigenfunctions considered for the estimation of the graphical model.

Omega

list of of precision matrices obtained using the multivariate functional principal component scores theta obtained using `fpca()`

Details

This function implements the functional graphical model in Zapata, Oh, and Petersen (2019). The arguments alpha and gamma are a reparameterization of the Group Graphical Lasso tuning parameters when using the JGL package. When using JGL::JGL, the tuning parameters are computed as lambda1 = alpha*gamma and lambda2 = (1-alpha)*gamma

References

Zapata J., Oh S. and Petersen A. (2019) - Partial Separability and Functional Graphical Models for Multivariate Gaussian Processes. Available at https://arxiv.org/abs/1910.03134.

Examples

# NOT RUN {
## Variables
# Omega - list of precision matrices, one per eigenfunction
# Sigma - list of covariance matrices, one per eigenfunction
# theta - list of functional  principal component scores
# phi - list of eigenfunctions densely observed on a time grid
# y - list containing densely observed multivariate (p-dimensional) functional data 

library(mvtnorm)
library(fda)

## Generate data y
 source(system.file("exec", "getOmegaSigma.R", package = "fgm"))
 theta = lapply(1:nbasis, function(b) t(rmvnorm(n = 100, sigma = Sigma[[b]])))
 theta.reshaped = lapply( 1:p, function(j){
     t(sapply(1:nbasis, function(i) theta[[i]][j,]))
 })
 phi.basis=create.fourier.basis(rangeval=c(0,1), nbasis=21, period=1)
 t = seq(0, 1, length.out = time.grid.length)
 chosen.basis = c(2, 3, 6, 7, 10, 11, 16, 17, 20, 21)
 phi = t(predict(phi.basis, t))[chosen.basis,]
 y = lapply(theta.reshaped, function(th) t(th)%*%phi)
 
## Solve
 fgm(y, alpha=0.5, gamma=0.8)
 

# }